Diquark Explanation of b → sℓ + ℓ −

The discrepancies between b → sℓ + ℓ − data and the corresponding Standard Model predictions point to the existence of new physics with a significance at the 5 σ level. While previously a lepton flavour universality violating effect was preferred, the new R ( K ( ∗ ) ) and B s → µ + µ − measurements are now compatible with the Standard Model, favouring a lepton flavour universal beyond the Standard Model contribution to C 9 . Since heavy new physics is generally chiral, and because of the stringent constraints from charged lepton flavour violation, this poses a challenge for model building. In this article, we point out a novel possibility: a diquark, i.e. a coloured scalar, induces the Wilson coefficient of the (¯ sγ µ P L b )(¯ cγ µ P L c ) operator at tree-level, which then mixes into O 9 via an off-shell photon penguin. This setup allows for a lepton flavour universal effect of C 9 ≈ − 0 . 5 , without violating bounds from ∆ M s , ∆Γ , B → X s γ and D 0 − ¯ D 0 mixing. This scenario predicts a small and negative C ′ 9 and a light diquark, preferably with a mass around 500 GeV, as compatible with the CMS di-di-jet analysis, and a deficit in the inclusive b → c ¯ cs rate.

Combining the processes discussed above in a global fit together with all other available data on b → sℓ + ℓ + transitions leads to a coherent picture.In fact, while before the R(K ( * ) ) update, the most strongly favoured scenarios were at least two-dimensional [49], now a single one-dimensional scenario is clearly favoured: the C U 9 scenario with a significance around 5σ [50][51][52][53].This means that a left-handed b − s current and a vectorial flavouruniversal lepton current (B s → µ + µ − [54-57] constrains an axial current) is needed.
An alternative scenario that can naturally generate C U 9 is a NP contribution to the Wilson coefficient of the (sγ µ P L b)(cγ µ P L,R c) operator [77] that mix into C 9 via an off-shell photon penguin [78].As a tree-level effect in sbcc operators is necessary, only Z ′ bosons, heavy gluons, Higgses [79][80][81][82] or diquarks (DQs) come into mind [83].
For the first two options the possible effect is stringently limited by B s − Bs mixing (to which these particles also contribute at tree-level [84]) and there is only a small region in parameter space left that works for 2HDMs [82].Therefore, we will consider the DQs in this article which have not been studied so far.Out of the 8 different scalar DQs [85,86], there is only a single representation that couples to left-handed down-type quarks, can have flavour-diagonal couplings to up-type quarks and does not lead to tree-level effects in ∆F = 2: the scalar SU (3) c triplet, SU (2) L singlet with hypercharge −1/3 which we call ϕ. 2DQs are not only theoretically well motivated, e.g. by E 6 models [87] or the R-parity violating MSSM [88], but also lead to interesting LHC signatures [89][90][91][92][93][94][95][96][97][98][99]: They are candidates from an explanation [100] of the ATLAS dijet [101] and the CMS di-di-jet [102] excesses.In fact, the non-resonant CMS analysis shows a weaker-thanexpected limit for a diquark SU (3) c triplet with a mass around 500 GeV.
In the next section, we will define our model and perform the matching on the effective theory before we continue with the phenomenological analysis in Sec.III can conclude in Sec.IV.

II. SETUP AND OBSERVABLES
There are 8 representations of scalar DQs with couplings to quarks SU (2) L singlets or doublets which can have either symmetric or anti-symmetric couplings in flavour space.In order to get a sizable effect in b → sℓ + ℓ − via the operator (sγ µ P L b)(cγ µ P L,R c), we need 1) simultaneous couplings to up-type and down-type quarks 2) flavour diagonal couplings to up-quarks (i.e.not antisymmetric ones) 3) left-handed down-quarks must be involved 4) no tree-level effect in B s − Bs mixing.These requirements only leave a single representation, i.e.IV in the conventions of Ref. [85] whose mass we label M ϕ .
The couplings to quarks are given by Here, α, β, γ are color indices, I, J (i, j) SU (2) L (flavour) indices and c stands for charge conjugation.Note that λL can be chosen to be symmetric in flavour space, i.e. λL ij = λL ji , without loss of generality.After electroweak symmetry breaking, the quark doublets decompose into their SU (2) L components and in the mass eigenbasis we have where we absorbed the rotation matrices into the definition of λ R and, working in the down basis, defined A. Tree-level Matching The Lagrangian with √ 2 and the operators defines the charged current interactions.Our diquark, integrated out at tree level at the electroweak scale, leads to the coefficients with the primed coefficients obtained by The effective Lagrangian governing b → sℓℓ transitions is given by 3 It can be seen that C uc X (which give b → ucs transitions) are enhanced relative to the SM by the large ratio V us /V ub .However, the effects of these operators in meson mixing-related observables is too small [103].Furthermore, C cu X (which give b → cūs transitions) have been considered as part of a potential explanation of the discrepancy between the SM prediction with QCD factorization and experiment for B0 [104][105][106][107][108][109][110][111].However as no analysis has been done with just NP in sbcu, this interesting future direction is beyond the scope of this article.
where the primed operators and coefficients are again obtained by exchanging L and R. The tree-level induced operators Q cc 1,2 generate via mixing at the B meson scale (and similarly for C ′ 9 ) with , and z = 4m 2 c /q 2 .Note that h also includes finite subleading q 2 -dependent effects (which we evaluate at q 2 = 5 GeV 2 ) [77].
The threshold effects from top quark loops induce at the EW scale with f (x) = x(x − log x − 1)/(x − 1) 2 and C ′ 9,10 obtained from C 9,10 by exchanging L and R.

C. B → Xsγ
The relevant effective operators are defined by with primed operators obtained through P R → P L .We have two NP contributions to C 7γ .The mixing of the sbcc operators into C 7γ leads to [77,112] where, similarly to the case of C 9 , we have included sub-leading q 2 dependent terms via ))/6.The equivalent result for the primed operators is again obtained by an exchange of chiralities. 4econd, the direct matching at the new physics scale (to be taken the weak scale) gives direct contributions to C (′) 7γ,8g (M W ). The most important result is the existence of m t /m b enhanced terms for both coefficients which are proportional to λ L 22 λ R 33 , while the full expressions can be found in the supplementary material [113].These coefficients are then evolved to the B meson scale resulting in The latest SM prediction for the inclusive radiative decay B → X s γ [114] Br[B → X s γ] SM = (3.40 ± 0.17) is in good agreement with the experimental average [115] Br This measurement stringently constrains a BSM contribution to C 7γ .In addition, asymmetries in B → K * e + e − [116] are very sensitive to C ′ 7γ .We performed a fit to C 7γ and C ′ 7γ using smelli [117][118][119].

D. Bs − Bs Mixing
In the SM, EW box diagrams give rise to B s − Bs mixing, which can be measured through observables including ∆M s , ∆Γ s , and a s sl that can in principle constrain our NP model.
The width difference for B s mesons can be calculated as )), where the current SM prediction is [130] and the latest HFLAV average5 is [115] ∆Γ s = (0.083 ± 0.005) ps −1 . ( Our model alters this quantity through the sbcc operators, where the full NP contributions were calculated in Refs.[77,112]. In addition the semi-leptonic asymmetry a s sl receives a large NP contribution since our NP does not suffer the severe GIM cancellation seen in the SM.However, at least orders of magnitude improvement in the experimental precision would be required for this effect to be observable, and so we make no further mention of it here (some more detailed discussion can be found in the supplementary material [113]).
The ∆C = 2 coefficients which give a short-distance contribution to ∆M D are The SM prediction is currently unclear, as a naive calculation gives a result four or five orders of magnitude too small, while other estimates (albeit not from first principles) suggest the SM alone could give a result x SM ∼ 0.1 % [131][132][133][134][135][136][137], where is the commonly reported observable in D 0 − D0 -mixing.
Comparing this to the current HLFAV average [115] x EXP = (0.407 ± 0.044) %, we take a conservative approach and allow the shortdistance NP contribution to be up to twice the size of the experimental value (i.e.we impose ∆M NP D ≤ 2∆M exp D , allowing in principle for a 100 % cancellation between BSM and SM).

F. Bs/B d lifetime ratio
The ratio of the B s to B d lifetimes was long thought to be a theoretically clean observable, benefiting from many cancellations of uncertainties.However, recent calculations of the SM contribution to the Darwin operator [138][139][140] has lead to a situation where the theory prediction is unclear, and in addition there is some tension between the different experimental measurements [115] (see the supplementary material [113] for further discussion).As such we do not consider this observable further.

III. PHENOMENOLOGY
We now turn to the phenomenology of our model.First of all, we set M ϕ = 500 GeV which is compatible with the non-resonant paired di-jet search of CMS due to the weaker-than-expected limit in this mass region [102].Note that using a light DQ helps to reduce the relative effect in ∆F = 2 processes since here the DQ contribution is proportional to λ 4 /M 2 ϕ while for all other flavour processes the leading DQ effect has a λ 2 /M 2 ϕ scaling.The product of λL 23 and λL 22 is necessary to give the effect in b → sℓ + ℓ − via C 9 while the product of λR 22 and λR 23 (λ R 12 ) helps to weaken the bound from B s − Bs mixing (D 0 − D0 ).To avoid a chirally enhanced effect in b → sγ λ R 33 ≈ 0 is helpful.Furthermore, to avoid bounds from di-jet resonance searches [141,142] and Kaon mixing, we assume the left-handed coupling involving the first generation to be approximately zero.Thus we consider the following structure for the DQ quark couplings Note that we input λL in the down quark basis (not including CKM rotations).Since λL * 22 λL 23 must be positive to give the preferred sign in C 9 we set for simplicity λL 22 = λL 23 and λ R 22 = λ R 23 , assume real couplings and show the preferred regions of the various observables in Fig. 1.We see that a reasonably sizable LFU C 9 , of the order of −0.5, can be generated in our model while still respecting the other experimental constraints.For D 0 − D0 -mixing, we show the regions compatible with our fine-tuning argument for λ R 12 = 0 and floating it within two different (small) ranges which are compatible with LHC di-jet searches.Interestingly, our model predicts |C ′ 9 | ≪ |C 9 | but with the same sign as (slightly) preferred by the current global fit [52].Note that generating the sizeable negative C 9 leads to a positive shift in ∆Γ s .This is more in line with the latest experimental result from CMS, which is in slight tension with LHCb and ATLAS measurements [115].

IV. CONCLUSIONS AND OUTLOOK
There are persistent and significant tensions in b → sℓ + ℓ − observables.
They are most pronounced in Br[B → Kµ + µ − ], in the angular observable P ′ 5 (in B → K * µ + µ − ) and the total branching ratio as well as angular observables in B s → ϕµ + µ − .In combination with the constraints from R(K ( * ) ) and B s → µ + µ − they point towards lepton flavour universal NP in C 9 with a significance at the 5σ level.This poses a challenge for model building because heavy NP is generally chiral, while for generating a dominant effect in C 9 , a vectorial lepton current is needed.Furthermore, bounds from charged lepton flavour violation require a separation of the electron and muon sectors, which is difficult to achieve in many models.
In this article, we proposed a novel model explaining the b → sℓ + ℓ − anomalies: an SU (3) c triplet scalar DQ with hypercharge Y = −1/3.This field can generate a LFU effect in C 9 via the mixing of the (sγ µ P L b)(cγ µ P L c) operator into O 9 .Since only quarks are directly involved in this model, charged lepton flavour violation is automatically respected.Furthermore, due to the weakerthan-expected LHC limit, DQs can still be relatively light, around 500 GeV.
Finally, our NP contributions to the C cc 1,2 coefficients have opposite sign than C cc 1 in the SM.This reduces the theory prediction for the inclusive branching ratio b → ccs where in the SM we have Br[b → ccs] SM = 23 ± 2% [149].However, while the situation for the experimental determination of this fully inclusive quantity is currently quite unclear, and O(30%) effects in Br[b → ccs] are possible, our model prediction is in line with the so-called "missing charm puzzle" [150].A clarification of the experimental situation would therefore be of great interest.

14 FIG. 1 .
FIG.1.Experimental constraints given our assumed UV couplings matrices.The black and red vertical and horizontal lines show the generated LFU contributions to C9 and C ′ 9 , respectively.